Approximation Theory, Sequence Spaces and Applications PDF
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Industrial and Applied Mathematics S. A. Mohiuddine Bipan Hazarika Hemant Kumar Nashine Editors Approximation Theory, Sequence Spaces and Applications Industrial and Applied Mathematics Editors-in-Chief G.D.VeerappaGowda,TIFRCentreForApplicableMathematics,Bengaluru, Karnataka,India S.Kesavan,InstituteofMathematicalSciences,Chennai,TamilNadu,India FahimaNekka,UniversitedeMontreal,Montréal,QC,Canada EditorialBoard AkhtarA.Khan,RochesterInstituteofTechnology,Rochester,USA GovindanRangarajan,IndianInstituteofScience,Bengaluru,India K.Balachandran,BharathiarUniversity,Coimbatore,TamilNadu,India K.R.Sreenivasan,NYUTandonSchoolofEngineering,Brooklyn,USA MartinBrokate,TechnicalUniversity,Munich,Germany M.ZuhairNashed,UniversityofCentralFlorida,Orlando,USA N.K.Gupta,IndianInstituteofTechnologyDelhi,NewDelhi,India NooreZahra,PrincessNourahbintAbdulrahmanUniversity,Riyadh,SaudiArabia PammyManchanda,GuruNanakDevUniversity,Amritsar,India RenéPierreLozi,UniversityCôted’Azur,Nice,France ZaferAslan,˙IstanbulAydınUniversity,˙Istanbul,Turkey TheIndustrialandAppliedMathematicsseriespublisheshigh-qualityresearch- levelmonographs,lecturenotes,textbooks,contributedvolumes,focusingonareas where mathematics is used in a fundamental way, such as industrial mathematics, bio-mathematics,financialmathematics,appliedstatistics,operationsresearchand computerscience. · · S. A. Mohiuddine Bipan Hazarika Hemant Kumar Nashine Editors Approximation Theory, Sequence Spaces and Applications Editors S.A.Mohiuddine BipanHazarika DepartmentofGeneralRequired DepartmentofMathematics Courses,Mathematics GauhatiUniversity TheAppliedCollege Guwahati,Assam,India KingAbdulazizUniversity Jeddah,SaudiArabia OperatorTheoryandApplicationsResearch Group,DepartmentofMathematics FacultyofScience KingAbdulazizUniversity Jeddah,SaudiArabia HemantKumarNashine MathematicsDivision,SchoolofAdvanced SciencesandLanguages VITBhopalUniversity Bhopal,MadhyaPradesh,India ISSN 2364-6837 ISSN 2364-6845 (electronic) IndustrialandAppliedMathematics ISBN 978-981-19-6115-1 ISBN 978-981-19-6116-8 (eBook) https://doi.org/10.1007/978-981-19-6116-8 MathematicsSubjectClassification:41-XX,46A45,46B45 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SingaporePteLtd.2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface This book features original chapters on the theory of approximation by positive linearoperatorsaswellastheoryofsequencespacesandillustratetheirinvolvement invariousapplications.Eachchapterdescribestheproblemofcurrentimportance, andtosummarizewaysoftheirsolutionandpossibleapplications,andtoimprove thecurrentunderstandingpertainingtosequencespacesandapproximationtheory. Thebookconsistsoforiginalfourteenresearchchapters.Corroborationsofthemain resultsaredetailedandelegant.Thelistofchaptersisarrangedalphabeticallybythe lastnamesofthefirstauthorofeachchapter. Chapter1isdevotedtostudyingthegeometricsequencespaceswithgeometric topologywhichisanonlineartopologyandexploressalientfeaturesofthegeometric topological sequence spaces. This chapter also discusses about normal topology and its properties, and introduces perfect, simple, symmetric geometric sequence spaces as well as duality between perfect geometric sequence spaces. Chapter 2 presents some results regarding the Fredholmnesss, compactness, and boundednes of composition operators defined on the second-order Cesáro function spaces and computesthecompositionoperators’sessentialnorm. Chapter 3 introduces the concepts of deferred statistical convergence of order αβ and strongly s-deferred Cesáro summability of order αβ of complex (or real) sequences and proves some related results. Chapter 4 obtains the asymptotic type theoremsuchastheVoronovskayatheorem,quantitativeandGrüss–Voronovskaya type theorem. The order of approximation for the functions having a derivative equivalent with a function of bounded variation for these operators is presented. Chapter5introducestheZacharyspaceoverR∞andfindsthatthisisaBanachspace offunctionsofboundedmeanoscillationwithorder p(1≤ p ≤∞)containingthe functionofboundedmeanoscillationasadensecontinuousembedding. Chapter 6 discusses some properties of Dunkl generalization of Szász opera- tors via q-calculus by considering the new generalization of the power summa- bilitymethodssuchasuniformconvergenceofthistypeofoperators,Korovkintype theorem,VoronovskayaandGrüss–Voronovskayatypetheorems,andobtainssome resultsonweightedspacesofcontinuousfunctions.Thischapterprovessomeresults relatedtothestatisticalconvergenceoftheDunklgeneralizationofSzászoperators v vi Preface viaq-calculusbyusingthe A-transformation.Moreover,therateofconvergenceof DunklgeneralizationofSzászoperatorsviaq-calculusforfunctionswithderivatives ofboundedvariationisestimated. Chapter7isdevotedtoconstructingtheSzász–Jakimovski–Leviatantypeopera- torsbyusingsequenceofnonnegativecontinuousfunctionsχ (z)on[0,∞).This n chapterinvestigatestheapproximationpropertiesofournewconstructedoperators involvingtheAppellpolynomialbyuseofmodulusofcontinuity,Lipschitzfunctions, Peetres K-functionalandweightedfunctions.Moreover,aninterestingapproxima- tionresultinthesenseof A-statisticalconvergenceispresented.Chapter8obtains theapproximationofsignalsbelongingtoZygmundclassassociatedwithconjugate Fourier series and conjugate derived Fourier series by using (E,r)A-mean which is the best approximation in comparison to the approximation obtained by using LipschitzclassandHölderspace. InChapter9,someeffectivesemi-analyticandnumericalmethodsarepresented basedonthefollowingapproaches.Thefirstapproachisamodifiedtechniquedevel- oped based on a concept of topology and perturbations theory which is named as modified homotopy perturbation method. The second method has been suggested based on Sinc function with suitable interpolation. The mentioned methods are discussed in detail and evaluated by obtaining the solutions for some nonlinear functional and fractional equations. These equations include integral and differen- tial equations in Banach spaces. The numerical solutions are introduced by some convergent iterative algorithms and computational operations are done by using Mathematica software. Moreover, the numerical results of the proposed methods are compared with some other ones found in literature which are developed based onsomebasisandorthogonalfunctions. Chapter10discussestheconstructionofanewsequenceofSzász-typeoperators involving q-Appell polynomials and gives some basic results for the new opera- tors.ThelocalapproximationresultsviaPeetre’sK-functional,Lipschitzclass,and modulus of smoothness are presented. This chapter also deals with the study of weightedapproximationresults,andstatisticalapproximationresultsarediscussed forthenewoperators.Chapter11introducessomeclassesofoperatorswhocommute with the Hilbert operator and obtain the (cid:2) -norm of two of those operators. This p chapteralsodeals,asanapplication,tofindthenormofsomewell-knownoperators onthesequencespacesassociatedwithHilbert’scommutants. Chapter 12 is dedicated to presenting a comprehensive literature review on the sequenceofuncertainvariables,complexuncertainvariablesdefinedbyOrliczfunc- tion.Intherecentyears,ordinarysequenceshavebeenextended tonewtypesand these extensions have been used in uncertain environment too, different type of convergenceisoneofthemajordevelopmentinthisdirections.Thisliteraturereview alsoanalyzesthechronologicaldevelopmentoftheseextensionsandpresentsthese interpretationsonthefutureofuncertaintytheory.Chapter13examinestheUlam– Hyers stability results of the mixed type additive-quadratic functional equation in thesettingofintuitionisticrandomnormedspaces. Preface vii Chapter 14 defines the q-Euler difference sequence spaces eq((cid:3)) and eq((cid:3)) 0 c derived by composition of the q-Euler matrix and the difference matrix (cid:3) in the spacesc andc,respectively.TheSchauderbases,α-,β-,andγ-dualsarediscussed 0 forthenewsequencespaces.Moreover,certainclassesofmatrixmappingsfromthe spacese0q((cid:3))andecq((cid:3))toanyoneofthespace(cid:2)∞,c,c0,or(cid:2)1arecharacterized. We wish to express our gratitude to the authors who have contributed to this book. We would like to thank our family for moral support during the preparation of this book. Finally, we are also very thankful to Mr. Shamim Ahmad, Editor of MathematicsinSpringer,fortakinginterestinpublishingthisbook. Jeddah,SaudiArabia S.A.Mohiuddine Guwahati,India BipanHazarika Bhopal,India HemantKumarNashine Contents 1 TopologyonGeometricSequenceSpaces ........................ 1 KhirodBoruahandBipanHazarika 2 CompositionOperatorsonSecond-OrderCesàroFunction Spaces ....................................................... 21 MetinBas¸arır,SerkanDemiriz,andEmrahEvrenKara 3 GeneralizedDeferredStatisticalConvergence .................... 35 MikailEt,HacerSengulKandemir,andMuhammedCinar 4 ApproximationbyGeneralizedLupas¸-PaˇltaˇneaOperators ........ 53 ArunKajlaandJyoti 5 ZacharySpacesZ p[R∞]andSeparableBanachSpaces .......... 71 HemantaKalita,BipanHazarika,andMohsenRabbani 6 New Generalization of the Power Summability Methods forDunklGeneralizationofSzászOperatorsviaq-Calculus ....... 83 ValdeteLoku, NaimL.Braha, NoraMahloul, andMaríadelCarmenListán-García 7 ApproximationbyGeneralizedSzász–Jakimovski–Leviatan TypeOperators ............................................... 119 Md.NasiruzzamanandM.Mursaleen 8 OnApproximationofSignals ................................... 139 B.P.PadhyandP.Baliarsingh 9 NumericalSolutionforNonlinearProblems ..................... 163 MohsenRabbani 10 Szász-TypeOperatorsInvolvingq-AppellPolynomials ............ 187 MohdRaiz,NadeemRao,andVishnuNarayanMishra 11 CommutantsoftheInfiniteHilbertOperators ................... 203 HadiRoopaei ix x Contents 12 OnComplexUncertainSequencesDefinedbyOrliczFunction ..... 221 SangeetaSahaandBinodChandraTripathy 13 Ulam-HyersStabilityofMixedTypeFunctionalEquation Deriving From Additive and Quadratic Mappings inIntuitionisticRandomNormedSpaces ........................ 243 K.Tamilvanan,M.Kameswari,andB.Sripathy 14 AStudyonq-EulerDifferenceSequenceSpaces .................. 257 TajaYayingandS.A.Mohiuddine
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